Finite Group Databasebeta

← C2×#24921920 #36312176

Subgroups — C2×#24921920 #36312176

Order 21248.

6566 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24921920C2132yesmaximal, direct product factor
2C2×#19128455C212yesmaximal, complemented
3C2×#19128449C211yesmaximal, characteristic, complemented
4C2×#19128407C214yesmaximal, complemented
5C2×#19128404C214yesmaximal, complemented
6C2×#19128389C214yesmaximal, complemented
7C2×#19128310C214yesmaximal, complemented
8C2×#19128299C214yesmaximal, complemented
9C2×#19128299C214yesmaximal, complemented
10C2×#19128236C212yesmaximal, complemented
11C2×#125×D83C211yesmaximal, characteristic, complemented
12C2×#564×C83C211yesmaximal, characteristic, complemented
13#19128455C22132yescomplemented
14#19128449C22116yescomplemented
15#19128407C22164yescomplemented
16#19128404C22164yescomplemented
17#19128389C22164yescomplemented
18#19128310C22164yescomplemented
19#19128299C22164yescomplemented
20#19128299C22164yescomplemented
21C2×#14581791C2214yes—
22#19128236C22132yescomplemented
23C2×#14581787C2214yes—
24C2×#14581787C2214yes—
25C2×#14581787C2214yes—
26C2×#14581787C2214yes—
27C2×#14581786C2214yes—
28C2×#14581784C2211yescharacteristic
29C2×#14581783C2212yes—
30C2×#14581783C2214yes—
31C2×#14581782C2214yes—
32C2×#14581782C2214yes—
33#125×D83C22116yescomplemented
34C2×#14581780C2212yes—
35C2×#14581777C2214yes—
36C2×#14581777C2214yes—
37C2×#14581777C2214yes—
38C2×#14581777C2214yes—
39C2×#14581777C2214yes—
40C2×#14581772C2214yes—
41C2×#14581772C2214yes—
42C2×#14581772C2212yes—
43C2×#14581770C2214yes—
44C2×#14581770C2214yes—
45C2×#14581770C2214yes—
46C2×#14581770C2214yes—
47C2×#14581767C2212yes—
48C2×#14581767C2214yes—
49C2×#14581767C2214yes—
50C2×C4×Dic166C2214yes—
51C2×#14581757C2212yes—
52C2×C4×D332C2214yes—
53C2×C4×D332C2214yes—
54C2×#14581754C2214yes—
55C2×#14581754C2212yes—
56C2×#14581754C2214yes—
57C2×#14581750C2214yes—
58C2×#14581750C2214yes—
59C2×C4⋊C4×D83C2214yes—
60C2×C4⋊C4×D83C2212yes—
61C2×C4⋊C4×D83C2214yes—
62C2×C42×D83C2211yescharacteristic
63#564×C83C22116yescomplemented
64C2×#125×C83C2211yescharacteristic
65C2×#125×C83C2214yes—
66C2×#125×C83C2212yes—
67C2×#125×C83C2214yes—
68C2×#125×C83C2214yes—
69C22×Dic166C2312yes—
70C22×Dic166C2314yes—
71C22×D332C2314yes—
72C22×D332C2312yes—
73C22×D332C2314yes—
74C22×C4×D83C2311yescharacteristic
75C22×C4×D83C2312yes—
76C22×C4×D83C2314yes—
77C22×C4×D83C2314yes—
78C22×C4×D83C2314yes—
79C22×C4×D83C2314yes—
80#14581791C23132yes—
81#14581787C23132yes—
82#14581787C23132yes—
83#14581787C23132yes—
84#14581787C23132yes—
85#14581786C23132yes—
86#14581784C2318yes—
87#14581783C23116yes—
88#14581783C23132yes—
89#14581782C23132yes—
90#14581782C23132yes—
91#14581780C23116yes—
92#14581777C23132yes—
93#14581777C23132yes—
94#14581777C23132yes—
95#14581777C23132yes—
96#14581777C23132yes—
97#14581772C23132yes—
98#14581772C23132yes—
99#14581772C23116yes—
100#14581770C23132yes—
101#14581770C23132yes—
102#14581770C23132yes—
103#14581770C23132yes—
104#14581767C23116yes—
105#14581767C23132yes—
106#14581767C23132yes—
107C4×Dic166C23132yes—
108#14581757C23116yes—
109C4×D332C23132yes—
110C4×D332C23132yes—
111#14581754C23132yes—
112#14581754C23132yes—
113#14581754C23116yes—
114#14581750C23132yes—
115#14581750C23132yes—
116C2×C4⋊Dic83C2314yes—
117C2×C4⋊Dic83C2314yes—
118C2×C4⋊Dic83C2312yes—
119C2×C4⋊Dic83C2314yes—
120C2×Dic83⋊C4C2314yes—
121C2×Dic83⋊C4C2314yes—
122C2×Dic83⋊C4C2314yes—
123C2×Dic83⋊C4C2314yes—
124C2×Dic83⋊C4C2314yes—
125C2×Dic83⋊C4C2314yes—
126C4⋊C4×D83C23132yes—
127C4⋊C4×D83C23132yes—
128C4⋊C4×D83C23116yes—
129C2×(C2×D83)⋊C4C2314yes—
130C2×(C2×D83)⋊C4C2314yes—
131C2×(C2×D83)⋊C4C2314yes—
132C2×(C2×D83)⋊C4C2314yes—
133C2×(C2×D83)⋊C4C2314yes—
134C2×(C2×D83)⋊C4C2314yes—
135C2×(C2×D83)⋊C4C2314yes—
136C2×(C2×D83)⋊C4C2314yes—
137C2×(C2×D83)⋊C4C2314yes—
138C2×C4×Dic83C2311yescharacteristic
139C2×C4×Dic83C2312yes—
140C2×C4×Dic83C2314yes—
141C2×C4×Dic83C2314yes—
142C42×D83C2318yes—
143#125×C83C23132yes—
144#125×C83C23132yes—
145#125×C83C23132yes—
146#125×C83C2318yes—
147#125×C83C23116yes—
148C2×C4⋊C4×C83C2314yes—
149C2×C4⋊C4×C83C2314yes—
150C2×C4⋊C4×C83C2312yes—
151C2×C4⋊C4×C83C2314yes—
152C2×C4⋊C4×C83C2314yes—
153C2×C4⋊C4×C83C2312yes—
154C2×C4⋊C4×C83C2314yes—
155C2×C4⋊C4×C83C2314yes—
156C2×C4⋊C4×C83C2312yes—
157C2×C42×C83C2311yescharacteristic
158C2×C42×C83C2314yes—
159C23×D83C2414yes—
160C23×D83C2411yescharacteristic
161C2×Dic166—48no—
162C2×Dic166—44no—
163C2×Dic166C24116yes—
164C2×Dic166C2418yes—
165C2×D332—216no—
166C2×D332—48no—
167C2×D332—44no—
168C2×D332C24116yes—
169C2×D332C24116yes—
170C2×D332C2418yes—
171C22×Dic83C2412yes—
172C22×Dic83C2414yes—
173C22×Dic83C2414yes—
174C22×Dic83C2411yescharacteristic
175C2×C4×D83—24no—
176C2×C4×D83—48no—
177C2×C4×D83—44no—
178C2×C4×D83—48no—
179C2×C4×D83—48no—
180C2×C4×D83—48no—
181C2×C4×D83C2414yes—
182C2×C4×D83C24116yes—
183C2×C4×D83C24116yes—
184C2×C4×D83C2418yes—
185C2×C4×D83C24116yes—
186C2×C4×D83C24116yes—
187C4⋊Dic83C2418yes—
188C4⋊Dic83C24116yes—
189C4⋊Dic83C24116yes—
190C4⋊Dic83C24116yes—
191Dic83⋊C4C24116yes—
192Dic83⋊C4C24116yes—
193Dic83⋊C4C24116yes—
194Dic83⋊C4C24116yes—
195Dic83⋊C4C24116yes—
196Dic83⋊C4C24116yes—
197(C2×D83)⋊C4C24116yes—
198(C2×D83)⋊C4C24116yes—
199(C2×D83)⋊C4C24116yes—
200(C2×D83)⋊C4C24116yes—
201(C2×D83)⋊C4C24116yes—
202(C2×D83)⋊C4C24116yes—
203(C2×D83)⋊C4C24116yes—
204(C2×D83)⋊C4C24116yes—
205(C2×D83)⋊C4C24116yes—
206C4×Dic83C2418yes—
207C4×Dic83C24116yes—
208C4×Dic83C2414yes—
209C4×Dic83C24116yes—
210C22×C4×C83C2414yes—
211C22×C4×C83C2414yes—
212C22×C4×C83C2412yes—
213C22×C4×C83C2411yescharacteristic
214C22×C4×C83C2414yes—
215C4⋊C4×C83C24116yes—
216C4⋊C4×C83C24116yes—
217C4⋊C4×C83C2418yes—
218C4⋊C4×C83C2418yes—
219C4⋊C4×C83C24116yes—
220C4⋊C4×C83C24116yes—
221C4⋊C4×C83C24116yes—
222C4⋊C4×C83C2418yes—
223C4⋊C4×C83C24116yes—
224C42×C83C2414yes—
225C42×C83C24116yes—
226C23×C83C2511yescharacteristic
227C22×D83—24no—
228C22×D83—28no—
229C22×D83—28no—
230C22×D83—28no—
231C22×D832−1+414yes—
232C22×D83C2518yes—
233C22×D83C2512yes—
234C2×C4×C83—22no—
235C2×C4×C83—28no—
236C2×C4×C83—28no—
237C2×C4×C83—28no—
238C2×C4×C83—24no—
239C2×C4×C83C2518yes—
240C2×C4×C83C2512yes—
241C2×C4×C83C2514yes—
242C2×C4×C83C2518yes—
243C2×C4×C83C2518yes—
244C2×Dic83—22no—
245C2×Dic83—24no—
246C2×Dic83—28no—
247C2×Dic83—28no—
248C2×Dic83C2514yes—
249C2×Dic83C2512yes—
250C2×Dic83C2518yes—
251C2×Dic83C2518yes—
252D332—232no—
253D332—48no—
254D332—416no—
255C4×D83—28no—
256C4×D83—416no—
257C4×D83—416no—
258C4×D83—416no—
259C4×D83—416no—
260C4×D83—48no—
261Dic166—48no—
262Dic166—416no—
263C22×C83C2×2−1+412yes—
264C22×C83#58514yes—
265C22×C83C2611yescharacteristic
266C2×D83—24no—
267C2×D83—28no—
268C2×D83—28no—
269C2×D83—28no—
270C2×D83—28no—
271C2×D83—416no—
272C2×D83C2×2−1+414yes—
273C4×C83—22no—
274C4×C83—28no—
275C4×C83—28no—
276C4×C83—24no—
277C4×C83—28no—
278Dic83—22no—
279Dic83—28no—
280Dic83—28no—
281Dic83—24no—
282C2×#3553—831nomaximal, Sylow
283C2×C83#355314yescomplemented
284C2×C83C22×2−1+411yescharacteristic
285C2×C83C2×#58512yes—
286D83—24no—
287D83—48no—
288#3553—8332no—
289C2×#564D8311yescharacteristic, complemented
290C2×#563—832no—
291C2×#562—831no—
292C2×#554—834no—
293C2×#553—834no—
294C2×#552—834no—
295C2×#541—834no—
296C2×#539—834no—
297C2×#539—834no—
298C2×#529—832no—
299C22×#125—831no—
300C83C2×#355311yescharacteristic, complemented, Sylow
301#564C2×D83116yescomplemented
302#563—8332no—
303#562—8316no—
304#554—8364no—
305#553—8364no—
306#552—8364no—
307#541—8364no—
308#539—8364no—
309#539—8364no—
310C2×C4⋊Q8—834no—
311#529—8332no—
312C2×#126—834no—
313C2×#126—834no—
314C2×#126—834no—
315C2×#126—834no—
316C2×#126—834no—
317C2×#125—8316no—
318C2×#125—834no—
319C2×#125—832no—
320C2×#125—834no—
321C2×#125—834no—
322C2×#125—831no—
323C2×#125C2×D8311yescharacteristic
324C2×#125C2×D8312yes—
325C2×#125C2×D8314yes—
326C2×#125C2×D8314yes—
327C2×#125C2×D8314yes—
328C2×#124—832no—
329C2×#123—834no—
330C2×#123—834no—
331C2×#123—834no—
332C2×#123—834no—
333C2×#123—834no—
334C2×C22⋊Q8—832no—
335C2×C22⋊Q8—834no—
336C2×C22⋊Q8—834no—
337C2×C22⋊Q8—834no—
338C2×C22⋊Q8—834no—
339C2×C22⋊Q8—834no—
340C2×C22⋊Q8—834no—
341C2×#121—832no—
342C2×#121—834no—
343C2×#121—834no—
344C2×C4×Q8—832no—
345C2×C4×Q8—834no—
346C2×C4×D4—832no—
347C2×C4×D4—834no—
348C2×C4×D4—834no—
349C2×C4×D4—834no—
350C2×C4×D4—834no—
351C2×#117—834no—
352C2×#117—834no—
353C22×C4⋊C4—832no—
354C22×C4⋊C4—834no—
355C22×C4⋊C4—834no—
356C22×C42—831no—
357C22×Q8—834no—
358C22×Q8—832no—
359C22×D4—834no—
360C22×D4—832no—
361C22×D4—834no—
362C23×C4—834no—
363C23×C4—832no—
364C23×C4—834no—
365C23×C4—834no—
366C23×C4—834no—
367C23×C4—831no—
368C4⋊Q8—8332no—
369#126—8332no—
370#126—8332no—
371#126—8332no—
372#126—8332no—
373#126—8332no—
374#125—8332no—
375#125—8316no—
376#125—8332no—
377#125—8332no—
378#125—838no—
379#125C22×D8318yes—
380#125C22×D83116yes—
381#125C22×D83132yes—
382#125C22×D83132yes—
383#125C22×D83132yes—
384#124—8316no—
385#123—8332no—
386#123—8332no—
387#123—8332no—
388#123—8332no—
389#123—8332no—
390C22⋊Q8—8332no—
391C22⋊Q8—8332no—
392C22⋊Q8—8332no—
393C22⋊Q8—8332no—
394C22⋊Q8—8316no—
395C22⋊Q8—8332no—
396C22⋊Q8—8332no—
397#121—8316no—
398#121—8332no—
399#121—8332no—
400C4×Q8—8316no—
401C4×Q8—8332no—
402C4×D4—8316no—
403C4×D4—8332no—
404C4×D4—8332no—
405C4×D4—8332no—
406C4×D4—8332no—
407#117—8332no—
408#117—8332no—
409C2×C4⋊C4—834no—
410C2×C4⋊C4—834no—
411C2×C4⋊C4—834no—
412C2×C4⋊C4—832no—
413C2×C4⋊C4—834no—
414C2×C4⋊C4—834no—
415C2×C4⋊C4—834no—
416C2×C4⋊C4—834no—
417C2×C4⋊C4—834no—
418C2×C4⋊C4—834no—
419C2×C4⋊C4—8316no—
420C2×C4⋊C4—8332no—
421C2×C4⋊C4—8332no—
422C2×C4⋊C4C22×D8314yes—
423C2×C4⋊C4C22×D8314yes—
424C2×C4⋊C4C22×D8312yes—
425C2×C4⋊C4C22×D8314yes—
426C2×C4⋊C4C22×D8314yes—
427C2×C4⋊C4C22×D8312yes—
428C2×C4⋊C4C22×D8314yes—
429C2×C4⋊C4C22×D8314yes—
430C2×C4⋊C4C22×D8312yes—
431C2×C22⋊C4—834no—
432C2×C22⋊C4—834no—
433C2×C22⋊C4—834no—
434C2×C22⋊C4—834no—
435C2×C22⋊C4—834no—
436C2×C22⋊C4—834no—
437C2×C22⋊C4—834no—
438C2×C22⋊C4—834no—
439C2×C22⋊C4—834no—
440C2×C42—831no—
441C2×C42—834no—
442C2×C42—834no—
443C2×C42—832no—
444C2×C42—838no—
445C2×C42C22×D8314yes—
446C2×C42C22×D8311yescharacteristic
447C24—831no—
448C24—834no—
449C2×Q8—3328no—
450C2×Q8—3324no—
451C2×Q8—8316no—
452C2×Q8—838no—
453C2×D4—3328no—
454C2×D4—3324no—
455C2×D4—8316no—
456C2×D4—838no—
457C2×D4—8316no—
458C2×D4—16616no—
459C22×C4—3328no—
460C22×C4—3328no—
461C22×C4—3328no—
462C22×C4—3328no—
463C22×C4—3324no—
464C22×C4—838no—
465C22×C4—8316no—
466C22×C4—8316no—
467C22×C4—8316no—
468C22×C4—8316no—
469C22×C4—832no—
470C22×C4—831no—
471C22×C4—834no—
472C22×C4—834no—
473C22×C4—834no—
474C22×C4—1664no—
475C22×C4C23×D8314yes—
476C22×C4C23×D8311yescharacteristic
477C22×C4C23×D8312yes—
478C22×C4C23×D8314yes—
479C22×C4C23×D8314yes—
480C4⋊C4—8316no—
481C4⋊C4—8316no—
482C4⋊C4—8316no—
483C4⋊C4—838no—
484C4⋊C4—8316no—
485C4⋊C4—8316no—
486C4⋊C4—8316no—
487C4⋊C4—8316no—
488C4⋊C4—8316no—
489C4⋊C4—8316no—
490C4⋊C4C23×D83116yes—
491C4⋊C4C23×D83116yes—
492C4⋊C4C23×D8318yes—
493C4⋊C4C23×D83116yes—
494C4⋊C4C23×D83116yes—
495C4⋊C4C23×D8318yes—
496C4⋊C4C23×D83116yes—
497C4⋊C4C23×D83116yes—
498C4⋊C4C23×D8318yes—
499C22⋊C4—8316no—
500C22⋊C4—8316no—
501C22⋊C4—8316no—
502C22⋊C4—8316no—
503C22⋊C4—8316no—
504C22⋊C4—8316no—
505C22⋊C4—8316no—
506C22⋊C4—8316no—
507C22⋊C4—8316no—
508C42—834no—
509C42—838no—
510C42—8316no—
511C42—8316no—
512C42C23×D8314yes—
513C42C23×D83116yes—
514C23—834no—
515C23—832no—
516C23—838no—
517C23—1668no—
518C23—1668no—
519C23—1668no—
520C23—1664no—
521C23C24×D8311yescharacteristic
522Q8—33216no—
523Q8—3328no—
524D4—33216no—
525D4—3328no—
526D4—16632no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—33216no—
533C2×C4—33216no—
534C2×C4—33216no—
535C2×C4—33216no—
536C2×C4—3328no—
537C2×C4—834no—
538C2×C4—832no—
539C2×C4—838no—
540C2×C4—838no—
541C2×C4—1668no—
542C2×C4—1662no—
543C2×C4—1664no—
544C2×C4—1668no—
545C2×C4—1668no—
546C2×C4C24×D8318yes—
547C2×C4C24×D8312yes—
548C2×C4C24×D8314yes—
549C2×C4C24×D8318yes—
550C2×C4C24×D8318yes—
551C22—33216no—
552C22—834no—
553C22—1668no—
554C22—1668no—
555C22—1668no—
556C22—1668no—
557C22—1664no—
558C222−1+4×D8312yes—
559C22#1912853814yes—
560C22C25×D8311yescharacteristic
561C4—22no—
562C4—28no—
563C4—28no—
564C4—28no—
565C4—24no—
566C4—1662no—
567C4—1664no—
568C4—1668no—
569C4—1668no—
570C2—3328no—
571C2—1664no—
572C2#2492192014yesdirect product factor
573C2C2×2−1+4×D8311yescharacteristic
574C2C2×#1912853812yes—
subgroupG/H
centerC23C24×D83
commutator, cosocleC22×C83C26
FrattiniC22C25×D83
FittingC2×#564×C83C2
socleC23×C83C25
hypercenterC2×#564D83
hypocenterC83C2×#3553
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#564

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #24921920C2direct (N × H)
2. C2×#19128455C2semidirect
3. C2×#19128449C2semidirect
4. C2×#19128407C2semidirect
5. C2×#19128404C2semidirect
6. C2×#19128389C2semidirect
7. C2×#19128310C2semidirect
8. C2×#19128299C2semidirect
9. C2×#19128299C2semidirect
10. C2×#19128236C2semidirect
11. C2×#125×D83C2semidirect
12. C2×#564×C83C2semidirect
13. #19128455C22semidirect
14. #19128449C22semidirect
15. #19128407C22semidirect
16. #19128404C22semidirect
17. #19128389C22semidirect
18. #19128310C22semidirect
19. #19128299C22semidirect
20. #19128299C22semidirect
22. #19128236C22semidirect
33. #125×D83C22semidirect
63. #564×C83C22semidirect
283. C2×C83#3553semidirect
289. C2×#564D83semidirect
300. C83C2×#3553semidirect
301. #564C2×D83semidirect
572. C2#24921920direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#249219202
C2×#191284552
C2×#191284492
C2×#191284072
C2×#191284042
C2×#191283892
C2×#191283102
C2×#191282992
C2×#191282362
C2×#125×D832
C2×#564×C832
C2×#355383